The Importance of Learning to Make Assumptions

David Fortus

Weizmann Institute of Science

How I Spend My Research Time

IQWSTMoDeLSDGOACMLeSLearning ProgressionsRevisiting Tomorrow 98

Ill-and Well-Defined Problems

Differ in their degree of constraint

A range of ill-definedness

Most real-world problems are ill-defined to some degree

Constraints and Solution Space

Solution space size is a major determinant of problem complexity.

Limiting solution space by application of constraints

Ill-defined problems can be transformed into well-defined problems by adding constraints.

Constraints can be subjective. These constraints are assumptions that need to be verified.

Skillfulness

In order to solve an ill-defined problem one needs to select and apply appropriate subjective constraints, and solve the resultant well-defined problem.

Does skill at solving ill-defined problems in a given domain imply skill at solving well-defined problems in the same domain? What about the opposite?

Experts versus Novices

• Experts notice patterns of information that are not attended to by novices

• Experts have acquired a great deal of content knowledge

• Experts’ knowledge is organized in ways that reflect a deep understanding of their subject matter

• Experts' knowledge cannot be reduced to sets of isolated facts: it reflects contexts of applicability

• Experts’ knowledge is retrievable with little conscious effort

Experts versus Novices (Specific to Ill-Defined Problems)

• Expert designers use explicit problem decomposition strategies, breaking down complex problems into simpler ones by the use of constraining assumptions

• Expert creative writers and computer programmers will often deliberately treat a problem as ill-defined, increasing its difficulty and their creativity

School Science

Traditionally built around well-defined problems Real-world science is extremely ill-defined There have been many reasons for reforming

school science – meaningfulness, interest, authenticity…

Can traditional school science help students develop the skills needed in real-world science?

???

Does skill at solving ill-defined problems in Newtonian mechanics imply skill at solving well-defined problems in the same domain? What about the opposite?

8 Participants 3 professors

– A - Physics & sci ed, 30 years experience, taught high school physics

– B - Physics, teaches undergrad courses– C - Physics, teaches undergrad courses

D - 1 post-doc in sci ed, BA in physics, taught high school physics

E-H - 4 sci ed grad students with BAs in physics All but H taught high school physics Only A-C exposed to ill-defined problems in

physics

Instruments

4 questions in Newtonian mechanics, video-taped “think-aloud” sessions– 3 well-defined, 1 ill-defined– same content– A bit of novelty in one of the well-defined questions.

Interviews about difficulties encountered

Question #1

A person is pouring buckets of water into a cylindrical swimming pool with a diameter of 15m. In an hour, the water level of the pool rises by 2cm. Assuming the person fills the 10-liter bucket and pours its contents into the swimming pool at a constant rate, how many buckets of water does the person pour into the pool every minute?

Question #2

A ball with a mass of 800gr is dropped, hitting the

ground with a speed of 1.0m/s. The ball

rebounds at a speed of 0.8m/s. The impact

lasted 1/20th of a second. What was the average

net force the ball was subjected to during the

impact?

Question #3

A block is pushed in order to start it sliding down an inclined plane with an angle of 30˚. The coefficient of friction between the plane and the block is velocity dependent: it is equal to μk = 0.15v, where v

is the block’s speed in m/s relative to the plane. Assuming the plane is very long, what is the maximum speed relative to the plane that the block can attain?

Question #4

A company thinks that there is a market for ultra-light umbrellas. While developing these umbrellas, the design team was confronted with the following question: Is the force of the rain falling on an opened umbrella a force that needs to be taken into consideration in designing an umbrella? You have been hired as a consultant to this firm. Your task is to estimate the magnitude of this force. You are allowed to use anything that you feel may assist you. You must be able to justify your solution before the members of the design team.

UMinn

• Context-rich problems– Are relevant to the lives of the students

– Do not depend on students knowing a trick

– Cannot be solved in a single step

– The unknown variable is not always specified

– The problem statement may not provide all the information that is needed to solve the problem, or it may provide more information than is needed

– Assumptions may be needed to simplify the problem.

The Difficulty of Context-Rich Problems

Features that Increase Difficulty• Unfamiliar context• Hard to learn physics• Multiple approaches needed (e.g.,

Newton’s 2nd law and the conservation of energy)

• Multiple subparts• Unspecified target variable• Superfluous information provided• Insufficient information provided• Assumptions need to be made• Involves vector component• Involves trigonometric identities• Involves calculus

Rate of Difficulty• 0-1: Easy• 1-3: Medium• 3-4 (not including 1st

feature): Difficult

Difficulty of the 4 Questions

Q1 & Q2 – Easy

Q3 & Q4 – Difficult

Unfamiliar Context

Difficult Physics

Multiple Approaches

Multiple Subparts

Unspecified Target

Variable

Superfluous Information

Insufficient Information

Assumptions Needed

Vectors Trig Calculus

Q #1

Q #2

Q #3 + + +Q #4 + + + +

Initial Analysis Framework

Bransford and Stein’s IDEAL problem-solving model:

– Identify the problem

– Define and represent the problem

– Explore possible strategies

– Act on the strategies

– Look back and evaluate the effects of your activities

Revised Analysis Framework

Participant A’s Solution to Problem #3

Participant D’s Solution to Problem #3

Participant A’s Solution to Problem #4

Participant D’s Solution to Problem #4

Problems 1-3: General Patterns

Other than participant H, all solved problems 1-3.

Participants A, B, and C, spent significantly less time strategizing than the other participants.

None of the participants remembered seeing a terminal velocity problem in the context of an inclined plane.

More time was spent evaluating the answers to problem 1 than the other problems because it involved the conversion of units.

Problem 4: Results

Two steps involved in transforming problem 4 into a well-defined problem:

– Breaking the problem into 3 sub-problems - the rate at which rain falls, the speed at which it falls, and the force which each raindrop applies to the umbrella.

– Making assumptions about - the size of a raindrop, the terminal speed of rain, the shape of raindrops, and the type of collision between the raindrops and the umbrella.

Problem 4: General Patterns

Only participants A-C solved problem 4. Why?

Participants A-G were able to break the problem down into sub-problems.

Only participants A-C were able to make the simplifying assumptions.

Participants D-G envisioned how they would design experiments to get the needed values.

Problem 4: Difficulty

Much more time was spent strategizing in problem 4 than in problems 1-3.

In problem 4 there were extended periods of silence – the cognitive demand was too great for the participants to think and talk simultaneously.

I often needed to remind the participants to “think aloud.”

Needed to “straighten out their thoughts.”

Problem 4: Nonlinearity

The solutions to problems 1-3 were linear – seldom were earlier steps revisited.

The solutions to problem 4 were nonlinear, jumping back and forth between “strategize” and “develop an equation.”

Some participants jumped back and forth between “strategize” and “read”, looking for additional information – they knew they were missing something.

Transfer

Few noticed any connection between the problems.

In problems 1-3, nobody strayed beyond the context of the problem. In problem 4, images form and connections to other contexts were elicited: “I’m singing in the rain…” Some critiqued problem 4.

The solutions to problems 1 & 2 are examples of low-road transfer; those to problems 3 & 4 are examples of high-road transfer.

Discussion

Only the participants with experience working with ill-defined physics problems were able to make the assumptions needed to solve problem 4.

Only problem 4 elicited connections to prior experience, a crucial condition for new knowledge to be integrated into existing knowledge.

High-road transfer becomes low-road only after multiple opportunities for varied practice.

Conclusion

In order to prepare people to deal with real-world physics problems, we need to include many more ill-defined problems into the physics curriculum.

It is not clear whether physics curriculum should be structured around real-world problems or only include them.